A Novel Approach to Discontinuous Bond Percolation Transition

TL;DR

This paper presents a new bond percolation model on lattices with multiple channels per bond, demonstrating a discontinuous transition in all dimensions, including one dimension, driven by the ratio of open valves.

Contribution

It introduces a novel percolation procedure with multiple channels and analytically shows a discontinuous transition in all dimensions, including one dimension.

Findings

Discontinuous percolation transition occurs in all spatial dimensions.

Transition threshold depends on the ratio of open valves, $rac{ }{ }$.

Special discontinuous percolation in one dimension with size-dependent channels.

Abstract

We introduce a bond percolation procedure on a $D$-dimensional lattice where two neighbouring sites are connected by $N$ channels, each operated by valves at both ends. Out of a total of $N$, randomly chosen $n$ valves are open at every site. A bond is said to connect two sites if there is at least one channel between them, which has open valves at both ends. We show analytically that in all spatial dimensions, this system undergoes a discontinuous percolation transition in the $N\to \infty$ limit when $\gamma =\frac{\ln n}{\ln N}$ crosses a threshold. It must be emphasized that, in contrast to the ordinary percolation models, here the transition occurs even in one dimensional systems, albeit discontinuously. We also show that a special kind of discontinuous percolation occurs only in one dimension when $N$ depends on the system size.